Consider the Anderson Hamiltonian H V = κ∆ V + ξ(·) on the multidimensional lattice torus V increasing to the whole of lattice, where ξ(·) is an i.i.d. potential with distribution function F . For K = 1, 2, . . ., let ψ(·; λ K, V ) be the eigenfunction of H V associated with the Kth largest eigenvalue λ K, V , and let z K, V ∈ V be the coordinate of the Kth larger value ξ K, V of ξ(·) in V . It is well-known that if F satisfies the condition log − log(1 − F (t)) = o(t) and some additional conditions on regular variation and continuity at infinity, then ψ(·; λ K, V ) is (asymptotically) completely localized at the site z τ (K), V , as a localization centre for the eigenfunction for some (random) τ (K) = τ V (K) 1. In this paper, we study the asymptotic behavior in probability of the indices τ V (K) as V increases and K 1 is fixed. In particular, we show that if F satisfies the condition − log(1 − F (t)) = O(t 3 ) (resp., −t −3 log(1 − F (t)) → ∞) and additional regularity conditions at infinity, then τ V (K) = O(1) (resp., τ V (K) → ∞) with high probability. For Weibull's and double exponential types distributions, we obtain the first order expansion formulas for log τ V (K)
